Vol 10, No 4 (2019)

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GENERALIZED COUPLED FIXED POINT THEOREMS ON BIPOLAR METRIC SPACES

ANIMESH GUPTA* AND GANESH KUMAR SONI

Department of Mathematics,

Swami Vivekanand Govt. P. G. College, Narsinghpur, India.

(Received On: 24-01-19; Revised & Accepted On: 02-04-19)

ABSTRACT

I

2000 Mathematics Subject Classification: 46A80, 47H10, 54H25.

Key words and phrases: Bipolar metric space, coupled fixed point, completeness.

INTRODUCTION

Since the year 1922, Banachs contraction principle, when many authors saw that these fixed point theorems can be utilized to investigate existence and uniqueness of solutions of periodic boundary value problems, difierential equations and nonlinear integral equations, these theorems attracted their attention. And, they extended these theorems to various generalizations of metric spaces as cone, partial and modular, e.g. [1, 2, 4, 5, 7, 9, 11, 12].

In literature, the notion of coupled fixed point has been introduced by Guo and Lakshmikantham [6] in 1987. Afterward, Bhaskar and Lakshmikantham [3] introduced certain coupled fixed point theorems in partially ordered metric spaces. We express a series of definitions of some fundamental notions related to bipolar metric spaces.

Definition 1: A bipolar metric space is a triple (𝑋, 𝑌, 𝑑) such that 𝑋,𝑌 ≠ 𝜙, and 𝑑 ∶ 𝑋× 𝑌 → 𝑅+ is a function satisfying the properties

(B0) if 𝑑 (𝑥, 𝑦) = 0, then 𝑥=𝑦, (B1) if 𝑥=𝑦, then 𝑑 (𝑥, 𝑦) = 0,

(B2) if, 𝑦 ∈ 𝑋 ∩ 𝑌 , then 𝑑 (𝑥, 𝑦) =𝑑 (𝑦, 𝑥), (B3) 𝑑(𝑥1,𝑦2)≤ 𝑑(𝑥1, 𝑦1) +𝑑(𝑥2, 𝑦1) +𝑑(𝑥2, 𝑦2),

for all (𝑥, 𝑦), (𝑥₁, 𝑦₁), (𝑥₂, 𝑦₂) ∈ 𝑋× 𝑌, where 𝑅+ symbolises the set of all non-negative real numbers. Then 𝑑 is called a bipolar metric on the pair (𝑋, 𝑌 ).

Definition 2: Let (𝑋₁, 𝑌₁) and (𝑋₂, 𝑌₂) be pairs of sets and given a function ∶𝑋₁∪ 𝑌₁→ 𝑋₂∪ 𝑌₂. If 𝑓(𝑋₁)⊆ 𝑋₂ and

𝑓(𝑌1)⊆ 𝑌_2, we call f a covariant map from (𝑋1, 𝑌1) to (𝑋2, 𝑌2) and denote this with 𝑓 ∶ (𝑋1, 𝑌1)→ (𝑋2, 𝑌2). If 𝑓(𝑋1)⊆ 𝑌2 and 𝑓(𝑌1)⊆ 𝑋2, then we call 𝑓 a contravariant map from (𝑋1, 𝑌1) to (𝑋2, 𝑌2) and write 𝑓 ∶ (𝑋1, 𝑌1)→

→ (𝑋2, 𝑌2). In particular, if 𝑑1 and 𝑑2 are bipolar metrics on (𝑋_1, 𝑌_1) and (X_2, Y_2), respectively, we sometimes

use the notations 𝑓 ∶ (𝑋₁, 𝑌₁, 𝑑₁)→ (𝑋₂, 𝑌₂, 𝑑₂) and 𝑓 ∶ (𝑋₁, 𝑌₁, 𝑑₁)→ (𝑋₂, 𝑌₂, 𝑑₂).

Definition 3: Let (𝑋, 𝑌, 𝑑) be a bipolar metric space. A point 𝑢 ∈ 𝑋 ∪ 𝑌 is called a left point if 𝑢 ∈ 𝑋, a right point if

𝑢 ∈ 𝑌 and a central point if it is both left and right point. Similarly a sequence (𝑥𝑛) on the set X is called a left sequence and a sequence (𝑦_𝑛) on Y is called a right sequence. In a bipolar metric space, a left or a right sequence is called simply a sequence. A sequence (𝑢_𝑛) is said to be convergent to a point 𝑢, iff (𝑢_𝑛) is a left sequence, 𝑢is a right point and 𝑙𝑖𝑚𝑛→ ∞𝑑(𝑢𝑛, 𝑢) = 0, or (𝑢𝑛) is a right sequence, 𝑢 is a left point and 𝑙𝑖𝑚𝑛→∞𝑑(𝑢, 𝑢𝑛) = 0. A bisequence (𝑥_𝑛, 𝑦_𝑛) on (𝑋, 𝑌, 𝑑) is a sequence on the set𝑋× 𝑌. If the sequences (𝑥_𝑛) and (𝑦_𝑛) are convergent, then the bisequence (𝑥_𝑛, 𝑦_𝑛) is said to be convergent, and if (𝑥_𝑛) and (𝑦_𝑛) converge to a common point, then (𝑥_𝑛, 𝑦_𝑛) is called biconvergent. (𝑥_𝑛, 𝑦_𝑛) is a Cauchy bisequence, if 𝑙𝑖𝑚_𝑛,𝑚→ ∞𝑑(𝑥𝑛, 𝑦𝑚) = 0. In a bipolar metric space, every

convergent Cauchy bisequence is biconvergent. A bipolar metric space is called complete, if every Cauchy bisequence is convergent, hence biconvergent.

Definition 4: Let (𝑋₁, 𝑌₁, 𝑑₁) and (𝑋₂, 𝑌₂, 𝑑₂) be bipolar metric spaces.

(1) A map 𝑓 ∶ (𝑋₁, 𝑌₁, 𝑑₁)→ (𝑋₂, 𝑌₂, 𝑑₂) is called left-continuous at a point 𝑥₀∈ 𝑋₁, if for every𝜖> 0, there exists a 𝛿> 0 such that 𝑑₁ (𝑥₀, 𝑦) <𝛿 implies 𝑑2�𝑓 (𝑥₀), 𝑓 (𝑦)�<𝜖 all y ∈𝑌₁ .

(2) A map 𝑓 ∶ (𝑋₁, 𝑌₁, 𝑑₁)→(𝑋₂, 𝑌₂, 𝑑₂) is called right-continuous at a point 𝑦₀∈ 𝑌₁, if for every 𝜖> 0, there exists a 𝛿> 0 such that 𝑑₁(𝑥, 𝑦₀) <𝛿implies 𝑑2�𝑓 (𝑥), 𝑓 (𝑦₀)�<𝜖 for all𝑥 ∈ 𝑋₁.

(3) A map f is called continuous, if it is left-continuous at each point 𝑥 ∈ 𝑋₁ and right-continuous at each point

𝑦 ∈ 𝑌1.

(4) A contravariant map 𝑓 ∶ (𝑋₁,𝑌₁,𝑑₁)→→ (𝑋₂,𝑌₂,𝑑₂) is continuous if and only if it is continuous as a covariant map 𝑓 ∶ (𝑋₁,𝑌₁,𝑑₁)→�𝑌₂,𝑋₂, 𝑑̅₂�.

It can be seen from the Definition \ref(def4) that a covariant or a contravariant map 𝑓 from (𝑋₁,𝑌₁,𝑑₁) to (𝑋₂,𝑌₂,𝑑₂) is continuous if and only if (𝑢_𝑛)→ 𝑣 on (𝑋₁,𝑌₁,𝑑₁) implies �𝑓 (𝑢_𝑛)� → 𝑓 (𝑣) on (𝑋₂,𝑌₂,𝑑₂).

In this paper, we extend certain coupled fixed point theorems, which can be considered as generalization of Banach fixed point theorem, to bipolar metric spaces. Also, we obtain some results which are related to these theorems. Finally, we give an example which presents the applicability of our obtained results.

COUPLED FIXED POINT THEOREMS IN BIPOLAR METRIC SPACES

Let Δ denote the class of all functions 𝛽 ∶ [0,∞ )→ [0,1) which satisfy the following conditions,

𝛽 (𝑡𝑛)→ 1 implies 𝑡𝑛 = 0 .

The following are examples of some functions belonging to \Delta. (1) 𝛽(𝑡) = k for 𝑠,𝑡 ∈[0,∞ ), where k ∈ [0, 1).

(2) 𝛽(𝑡) = 𝑙𝑜𝑔 (1+𝑘𝑡)

𝑘𝑡 , 𝑡 > 0.

Now, we will prove our main result.

Definition 5: Let (𝑋,𝑌,𝑑) be a bipolar metric space, 𝐹 ∶(𝑋2,𝑌2)→ (𝑋,𝑌) be a covariant mapping. (𝑎,𝑏)∈ 𝑋2∪ 𝑌2 is said to be a coupled fixed point of F if 𝐹(𝑎,𝑏) =𝑎 and 𝐹(𝑏,𝑎) =𝑏.

Let Δ denote the class of those functions 𝛽 ∶ [0,∞ )→[0,1) which satisfy the following condition

𝛽 (𝑡𝑛)→ 1 implies 𝑡𝑛 = 0 .

Theorem 6: Let (𝑋,𝑌,𝑑) be a complete bipolar metric space, 𝐹 ∶(𝑋2,𝑌2)→ (𝑋,𝑌) be a covariant mapping and

𝛽1,𝛽2 ∈ Δ. If F satisfies the condition

𝑑�𝐹(𝑎,𝑏),𝐹(𝑝,𝑞)� ≤ 𝛽1 (𝑑(𝑎,𝑝))𝑑(𝑎,𝑝) +𝛽2(𝑑(𝑏,𝑞))𝑑(𝑏,𝑞), (1)

for all 𝑎,𝑏 ∈ 𝑋,𝑝,𝑞 ∈ 𝑌, then 𝐹 ∶ 𝑋2∪ 𝑌2→ 𝑋 ∪ 𝑌 has a unique coupled fixed point.

Proof: Let 𝑎0,𝑏0 ∈ 𝑋 and 𝑝0,𝑞0 ∈ 𝑌. We take 𝑎1,𝑏1 ∈ 𝑋 and 𝑝1,𝑞1 ∈ 𝑌

with 𝑎₁ = 𝐹(𝑎₀,𝑏₀),𝑏₁ = 𝐹(𝑏₀,𝑎₀),𝑝₁ =𝐹(𝑝₀,𝑞₀),𝑞₁ = 𝐹(𝑞₀,𝑝₀) .And similarly, we take 𝑎₂,𝑏₂∈ 𝑋 and

𝑝2,𝑞2 ∈ 𝑌with 𝑎2 = 𝐹(𝑎1,𝑏1),𝑏2 = 𝐹(𝑏1,𝑎1),𝑝2 = 𝐹(𝑝1,𝑞1),𝑞2 = 𝐹(𝑞1,𝑝1).

In this way, we obtain bisequences (𝑎𝑛,𝑏𝑛) and (𝑝𝑛,𝑞𝑛) with

𝑎𝑛+1 = 𝐹(𝑎𝑛,𝑏𝑛), 𝑏𝑛+1 = 𝐹(𝑏𝑛,𝑎𝑛), 𝑝𝑛+1 = 𝐹(𝑝𝑛,𝑞𝑛) and 𝑞𝑛+1 = 𝐹(𝑞𝑛,𝑝𝑛) for all 𝑛 ∈ 𝑁+. From (1), we get

𝑑(𝑎𝑛,𝑝𝑛+1) =𝑑�𝐹(𝑎𝑛−1,𝑏𝑛−1),𝐹(𝑝𝑛,𝑞𝑛)�,

≤ 𝛽1�𝑑(𝑎𝑛−1,𝑝𝑛)�𝑑(𝑎𝑛−1,𝑝𝑛) +𝛽2�𝑑(𝑏𝑛−1,𝑞𝑛)�𝑑(𝑏𝑛−1,𝑞𝑛) (2) and

𝑑(𝑏𝑛,𝑞𝑛+1) =𝑑�𝐹(𝑏𝑛−1,𝑎𝑛−1),𝐹(𝑞𝑛,𝑝𝑛)�,

≤ 𝛽1�𝑑(𝑏𝑛−1,𝑞𝑛)�𝑑(𝑏𝑛−1,𝑞𝑛) +𝛽2�𝑑(𝑎𝑛−1,𝑝𝑛)�𝑑(𝑎𝑛−1,𝑝𝑛) (3)

for all 𝑛 ∈ 𝑁+. Let

𝑒𝑛 = 𝑑(𝑎𝑛,𝑝𝑛+1) +𝑑(𝑏𝑛,𝑞𝑛+1)

for all𝑛 ∈ 𝑁+. Combining (2) and (3), we observe that

𝑒𝑛=𝑑(𝑎𝑛,𝑝𝑛+1) +𝑑(𝑏𝑛,𝑞𝑛+1)

≤ 𝛽1�𝑑(𝑎𝑛−1,𝑝𝑛)�𝑑(𝑎𝑛−1,𝑝𝑛) +𝛽2�𝑑(𝑏𝑛−1,𝑞𝑛)�𝑑(𝑏𝑛−1,𝑞𝑛)

+ 𝛽1�𝑑(𝑏𝑛−1,𝑞𝑛)�𝑑(𝑏𝑛−1,𝑞𝑛) + 𝛽2�𝑑(𝑎𝑛−1,𝑝𝑛)�𝑑(𝑎𝑛−1,𝑝𝑛)

Then we get

0≤ 𝑒𝑛≤ 𝑒𝑛−1 ≤ 𝑒𝑛−2 ≤ ⋯ 𝑒0. (4)

On the other hand,

𝑑(𝑎𝑛+1,𝑝𝑛) =𝑑�𝐹(𝑎𝑛,𝑏𝑛),𝐹(𝑝𝑛−1,𝑞𝑛−1)�,

≤ 𝛽_1�𝑑(𝑎𝑛,𝑝𝑛−1)�𝑑(𝑎𝑛,𝑝𝑛−1) +𝛽2�𝑑(𝑏𝑛,𝑞𝑛−1)�𝑑(𝑏𝑛,𝑞𝑛−1) (5)

and

𝑑(𝑏𝑛+1,𝑞𝑛) =𝑑�𝐹(𝑏𝑛,𝑎𝑛),𝐹(𝑞𝑛−1,𝑝𝑛−1)�,

≤ 𝛽1�𝑑(𝑏𝑛,𝑞𝑛−1)�𝑑(𝑏𝑛,𝑞𝑛−1) +𝛽2�𝑑(𝑎𝑛,𝑝𝑛−1)�𝑑(𝑎𝑛,𝑝𝑛−1) (6)

for all 𝑛 ∈ 𝑁+ and 𝜆< 1. Let

𝑠𝑛=𝑑(𝑎𝑛+1,𝑝𝑛) + 𝑑(𝑏𝑛+1,𝑞𝑛)

for all n ∈ N^+. Combining (5) and (6), we observe that

𝑠𝑛=𝑑(𝑎𝑛+1,𝑝𝑛) + 𝑑(𝑏𝑛+1,𝑞𝑛)

≤ 𝛽1�𝑑(𝑎𝑛,𝑝𝑛−1)�𝑑(𝑎𝑛,𝑝𝑛−1) +𝛽2�𝑑(𝑏𝑛,𝑞𝑛−1)�𝑑(𝑏𝑛,𝑞𝑛−1)

+ 𝛽1�𝑑(𝑏𝑛,𝑞𝑛−1)�𝑑(𝑏𝑛,𝑞𝑛−1) +𝛽2�𝑑(𝑎𝑛,𝑝𝑛−1)�𝑑(𝑎𝑛,𝑝𝑛−1)

= (𝛽1 + 𝛽2)(𝑑(𝑎𝑛,𝑝𝑛−1) + 𝑑(𝑏_𝑛,𝑞_(𝑛 −1)))(𝑑(𝑎𝑛,𝑝𝑛−1) + 𝑑(𝑏𝑛,𝑞𝑛−1)

=𝑠𝑛−1.

Then similar to Equation (4), we obtain that

0≤ 𝑠𝑛≤ 𝑠𝑛−1≤ 𝑠𝑛−2≤ ⋯ 𝑠0. (7)

Moreover,

𝑑(𝑎𝑛,𝑝𝑛) =𝑑�𝐹(𝑎𝑛−1,𝑏𝑛−1),𝐹(𝑝𝑛−1,𝑞𝑛−1)�,

≤ 𝛽1�𝑑(𝑎𝑛−1,𝑝𝑛−1)�𝑑(𝑎𝑛−1,𝑝𝑛−1) +𝛽2�𝑑(𝑏𝑛−1,𝑞𝑛−1)�𝑑(𝑏𝑛−1,𝑞𝑛−1) (8) and

𝑑(𝑏𝑛,𝑞𝑛) =𝑑�𝐹(𝑏𝑛−1,𝑎𝑛−1),𝐹(𝑞𝑛−1,𝑝𝑛−1)�,

≤ 𝛽1�𝑑(𝑏𝑛−1,𝑞𝑛−1)�𝑑(𝑏𝑛−1,𝑞𝑛−1) +𝛽_2�𝑑(𝑎𝑛−1,𝑝𝑛−1)�𝑑(𝑎𝑛−1,𝑝𝑛−1) (9)

for all 𝑛 ∈ 𝑁+. Therefore, let

𝑡𝑛 = 𝑑(𝑎𝑛,𝑝𝑛) + 𝑑(𝑏𝑛,𝑞𝑛)

for all 𝑛 ∈ 𝑁+ . Combining (8) and (9), we observe that

𝑡𝑛=𝑑(𝑎𝑛,𝑝𝑛) + 𝑑(𝑏𝑛,𝑞𝑛)

≤ 𝛽1�𝑑(𝑎𝑛−1,𝑝𝑛−1)�𝑑(𝑎𝑛−1,𝑝𝑛−1) +𝛽2�𝑑(𝑏𝑛−1,𝑞𝑛−1)�𝑑(𝑏𝑛−1,𝑞𝑛−1)

+𝛽1�𝑑(𝑏𝑛−1,𝑞𝑛−1)�𝑑(𝑏𝑛−1,𝑞𝑛−1) +𝛽2�𝑑(𝑎𝑛−1,𝑝𝑛−1)�𝑑(𝑎𝑛−1,𝑝𝑛−1)

= (𝛽1+𝛽2 )�𝑑(𝑎𝑛−1,𝑝𝑛−1) +𝑑(𝑏𝑛−1,𝑞𝑛−1)��𝑑(𝑎𝑛−1,𝑝𝑛−1) +𝑑(𝑏𝑛−1,𝑞𝑛−1)�

=𝑡𝑛−1.

Thus, we obtain that

0≤ 𝑡𝑛≤ 𝑡𝑛−1≤ 𝑡𝑛−2≤ ⋯ 𝑡0. (10)

Using the property (B3), we get

𝑑(𝑎𝑛,𝑝𝑚)≤ 𝑑(𝑎𝑛,𝑝𝑛+1) +𝑑(𝑎𝑛+1,𝑝𝑛+1) +⋯ + 𝑑(𝑎𝑚−1,𝑝𝑚),

𝑑(𝑏𝑛,𝑞𝑚)≤ 𝑑(𝑏𝑛,𝑞𝑛+1) +𝑑(𝑏𝑛+1,𝑞𝑛+1) +⋯ + 𝑑(𝑏𝑚−1,𝑞𝑚) (11)

and

𝑑(𝑎𝑚,𝑝𝑛)≤ 𝑑(𝑎𝑚,𝑝𝑚−1) +𝑑(𝑎𝑚−1,𝑝𝑚−1) +⋯ + 𝑑(𝑎𝑛+1,𝑝𝑛),

𝑑(𝑏𝑚,𝑞𝑛)≤ 𝑑(𝑏𝑚,𝑞𝑚−1) +𝑑(𝑏𝑚−1,𝑞𝑚−1) +⋯ + 𝑑(𝑏𝑛+1,𝑞𝑛) (12) for each 𝑚 ,𝑛 ∈ 𝑁 , 𝑛 < 𝑚 . Then, from (4), (7) , (10), (11) and (12), we have

𝑑(𝑎𝑛,𝑝𝑚) + 𝑑(𝑏𝑛,𝑞𝑚)≤ �𝑑(𝑎𝑛,𝑝𝑛+1) + 𝑑(𝑏𝑛,𝑞𝑛+1)�

+�𝑑(𝑎𝑛+1,𝑝𝑛+1) + 𝑑(𝑏𝑛+1,𝑞𝑛+1)�+⋯

+�𝑑(𝑎𝑚−1,𝑝𝑚−1) + 𝑑(𝑏𝑚−1,𝑞𝑚−1)�

+�𝑑(𝑎𝑚−1,𝑝𝑚) + 𝑑(𝑏𝑚−1,𝑞𝑚)�,

=𝑒𝑛+ 𝑡𝑛+1+ 𝑒𝑛+1 +⋯ + 𝑡𝑚−1 + 𝑒𝑚−1,

< 𝑒0+ 𝑡0 (13)

and

𝑑(𝑎𝑚,𝑝𝑛) + 𝑑(𝑏𝑚,𝑞𝑛)≤ �𝑑(𝑎𝑚,𝑝𝑚−1) +𝑑(𝑏𝑚,𝑞𝑚−1)�

+�𝑑(𝑎𝑚−1,𝑝𝑚−1) +𝑑(𝑏𝑚−1,𝑞𝑚−1)�+⋯

+�𝑑(𝑎𝑛+1,𝑝𝑛+1) +𝑑(𝑏𝑛+1,𝑞𝑛+1)�

+�𝑑(𝑎𝑛+1,𝑝𝑛) +𝑑(𝑏𝑛+1,𝑞𝑛)�,

=𝑠𝑛+ 𝑡𝑚−1+ 𝑡𝑚−1 +⋯ + 𝑡𝑛+1 + 𝑠𝑛+1,

for 𝑛 < 𝑚. Since, for an arbitrary 𝜖> 0, there exists n_0 such that (𝑒0+ 𝑡0) < 𝜖

3 and (𝑠0+ 𝑡0) < 𝜖

3 from (13)

and (14) we have

𝑑(𝑎𝑛,𝑝𝑚) +𝑑(𝑏𝑛,𝑞𝑚) <₃ 𝜖

for each 𝑛,𝑚 ≥ 𝑛₀. Then (𝑎_𝑛,𝑝_𝑛) and (𝑏_𝑛,𝑞_𝑛) are Cauchy bisequences. Because of completeness of (X, Y, d), there exist a, b∈ X and 𝑝,𝑞 ∈ 𝑌 with

𝑙𝑖𝑚𝑛→∞𝑎𝑛 = 𝑝 , 𝑙𝑖𝑚𝑛→∞𝑏𝑛=𝑞 ,𝑙𝑖𝑚𝑛→∞𝑝𝑛= 𝑎, 𝑎𝑛𝑑𝑙𝑖𝑚𝑛→∞𝑞𝑛=𝑏 . (15)

Then there exists 𝑛₁∈ 𝑁 with

𝑑(𝑎𝑛,𝑝) < 𝜖_{3 ,} 𝑑(𝑏𝑛,𝑞) <_{3 ,} 𝜖 𝑑(𝑎,𝑝𝑛) < _{3 ,}𝜖 𝑎𝑛𝑑 𝑑(𝑏,𝑞𝑛) < 𝜖₃

for all 𝑛 ≥ 𝑛1 and every ϵ > 0. Since (𝑎𝑛,𝑝𝑛) and (𝑏𝑛,𝑞𝑛) are Cauchy bisequences, we get

𝑑(𝑎𝑛,𝑝𝑛) < 𝜖₃ 𝑎𝑛𝑑 𝑑(𝑏𝑛,𝑞𝑛) <𝜖₃ So, from (1), we have

𝑑(𝐹(𝑎,𝑏),𝑝)≤ 𝑑(𝐹(𝑎,𝑏),𝑝𝑛+1) +𝑑(𝑎𝑛+1,𝑝𝑛+1) +𝑑(𝑎𝑛+1,𝑝)

=𝑑�𝐹(𝑎,𝑏),𝐹(𝑝𝑛,𝑞𝑛)� +𝑑(𝑎𝑛+1,𝑝𝑛+1) + 𝑑(𝑎𝑛+1,𝑝)

≤ 𝛽1�𝑑(𝑎,𝑝𝑛)�𝑑(𝑎,𝑝𝑛) +𝛽2𝑏�𝑑(𝑏,𝑞𝑛)�𝑑(𝑏,𝑞𝑛) +𝑑(𝑎𝑛+1,𝑝𝑛+1) +𝑑(𝑎𝑛+1,𝑝) <𝛽1�𝜖

3�

𝜖

3 + 𝛽1�

𝜖

3�

𝜖

3 +

𝜖

3 +

𝜖

3 =_{3 + 2}𝜖 𝜖₃

<𝜖

for each 𝑛 ∈ 𝑁 and 𝜆< 1. Then 𝑑(𝐹(𝑎,𝑏),𝑝) = 0. Hence, 𝐹(𝑎,𝑏) =𝑝. Similarly, we get

𝐹(𝑏,𝑎) = 𝑞, 𝐹(𝑝,𝑞) = 𝑎 𝑎𝑛𝑑 𝐹(𝑞,𝑝) =𝑏. On the other hand, from (15) we get

𝑑(𝑎,𝑝) =𝑑( 𝑙𝑖𝑚𝑛→∞𝑝𝑛,𝑙𝑖𝑚𝑛→∞ 𝑎𝑛) =𝑙𝑖𝑚𝑛→∞𝑑(𝑎𝑛,𝑝𝑛) = 0

and

𝑑(𝑏,𝑞) =𝑑(𝑙𝑖𝑚𝑛→∞𝑞𝑛, 𝑙𝑖𝑚𝑛→∞𝑏𝑛) =𝑙𝑖𝑚𝑛→∞𝑑(𝑏𝑛,𝑞𝑛) = 0.

So, 𝑎 = 𝑝 and 𝑏 = 𝑞. Therefore, (𝑎,𝑏)∈ 𝑋2∩ 𝑌2is a coupled fixed point of 𝐹.

Now, to show the uniqueness, we begin by taking another coupled fixed point (𝑎∗,𝑏∗)∈ 𝑋2∪ 𝑌2. If (𝑎∗,𝑏∗)∈ 𝑋2, then we get

𝑑(𝑎∗_{,𝑎) = 𝑑�𝐹(𝑎}∗_,𝑏∗_{),𝐹(𝑎,𝑏)� ≤ 𝑘𝑑(𝑎}∗ _{,𝑎) +𝑙𝑑(𝑏}∗ _,𝑏)

and

𝑑(𝑏∗ _{,𝑏) = 𝑑�𝐹(𝑏}∗ _,𝑎∗ _{),𝐹(𝑏,𝑎)� ≤ 𝑘𝑑(𝑏}∗_{,𝑏) +𝑙𝑑(𝑎}∗_,𝑎).

Therefore, we have

𝑑(𝑎∗_{,𝑎) +𝑑(𝑏}∗_{,𝑏)≤ 𝜆�𝑑(𝑎}∗_{,𝑎) +𝑑(𝑏}∗_{,𝑏)�. (16)}

Since 𝜆 < 1, by (16) this means that 𝑑(𝑎∗,𝑎) +𝑑(𝑏∗,𝑏) = 0. So, we obtain that 𝑎∗=𝑎 and 𝑏∗=𝑏. Similarly, if

(𝑎∗_,𝑏∗_{)∈ 𝑌}2_{, we have 𝑎}∗_{=𝑎 and 𝑏}∗_{=𝑏. Then (𝑎,𝑏) is a unique coupled fixed point of F.}

The following corollary is obtained, if we take 𝛽₁(𝑡) =𝑘 and 𝛽₂ (𝑡) =𝑙 in Theorem 6.

Corollary 7: Let (𝑋,𝑌,𝑑) be a complete bipolar metric space, 𝐹 ∶(𝑋2,𝑌2)→ (𝑋,𝑌 ) be a covariant mapping and k, l be non-negative constants. If F satisfies the condition

𝑑�𝐹(𝑎,𝑏),𝐹(𝑝,𝑞)� ≤ 𝑘𝑑(𝑎,𝑝) +𝑙𝑑(𝑏,𝑞), 𝑘+𝑙< 1 (17) for all 𝑎,𝑏 ∈ 𝑋,𝑝,𝑞 ∈ 𝑌, then 𝐹 ∶ 𝑋2∪ 𝑌2→ 𝑋 ∪ 𝑌 has a unique coupled fixed point.

The following corollary is obtained, if we take equal the constants k, l in Theorem 6.

Corollary 8: Let (𝑋, 𝑌, 𝑑) be a complete bipolar metric space, 𝐹 ∶ (𝑋2, 𝑌2)→(𝑋, 𝑌 ) be a covariant mapping and k, l be non-negative constants. If the condition

𝑑�𝐹(𝑎, 𝑏), 𝐹(𝑝, 𝑞)� ≤𝑘₂ ��𝑑(𝑎, 𝑝) +𝑑(𝑏, 𝑞)�,�

𝑘< 1 holds for all 𝑎, 𝑏 ∈ 𝑋,𝑝, 𝑞 ∈ 𝑌, then 𝐹 ∶ 𝑋2∪ 𝑌2→ 𝑋 ∪ 𝑌 has a unique coupled fixed point.

Now, we express another generalization of coupled fixed point theorem in bipolar metric spaces.

Definition 9: Let (𝑋, 𝑌, 𝑑) be a bipolar metric space, 𝑎 ∈ 𝑋,𝑝 ∈ 𝑌 and 𝐹 ∶ (𝑋 × 𝑌, 𝑌× 𝑋) → (𝑋, 𝑌) be a covariant mapping. (𝑎, 𝑝) is said to be a coupled fixed point of 𝐹 if 𝐹(𝑎, 𝑝) = 𝑎 and 𝐹(𝑝, 𝑎) = 𝑝.

Theorem 10: Let (𝑋, 𝑌, 𝑑) be a complete bipolar metric space, 𝐹 ∶ (𝑋× 𝑌, 𝑌 × 𝑋)→ (𝑋, 𝑌) be a covariant mapping and 𝛽₁ ,𝛽₂∈ Δ. If the condition

holds for all 𝑎, 𝑏 ∈ 𝑋,𝑝, 𝑞 ∈ 𝑌, then 𝐹 ∶ (𝑋× 𝑌 )∪ (𝑌× 𝑋)→ 𝑋 ∪ 𝑌 has a unique coupled fixed point.

Proof: Similar to the proof of Theorem 6, we define bisequences (𝑎_𝑛, 𝑝_𝑛) and (𝑏_𝑛, 𝑞_𝑛) as follows,

𝑎𝑛+1 = 𝐹(𝑎𝑛,𝑏𝑛),𝑏𝑛+1 = 𝐹(𝑏𝑛,𝑎𝑛), 𝑝𝑛+1= 𝐹(𝑝𝑛,𝑞𝑛) and 𝑞𝑛+1= 𝐹(𝑞𝑛,𝑝𝑛)

for all 𝑛 ∈ 𝑁+. Let 𝑘 + 𝑙 =𝜆. From \ref (18) , we get

𝑑(𝑎𝑛,𝑞𝑛+1) =𝑑�𝐹(𝑎𝑛−1,𝑝𝑛−1),𝐹(𝑞𝑛,𝑏𝑛)�,

≤ 𝛽1�𝑑(𝑎𝑛−1,𝑞𝑛)�𝑑(𝑎𝑛−1,𝑞𝑛) +𝛽2�𝑑(𝑏𝑛,𝑝𝑛−1)�𝑑(𝑏𝑛,𝑝𝑛−1) (19)

𝑑(𝑎𝑛+1,𝑞𝑛) =𝑑�𝐹(𝑎𝑛,𝑝𝑛),𝐹(𝑞𝑛−1,𝑏𝑛−1)�,

≤ 𝛽1�𝑑(𝑎𝑛,𝑞𝑛−1)�𝑑(𝑎𝑛,𝑞𝑛−1) +𝛽2�𝑑(𝑏𝑛−1,𝑝𝑛)�𝑑(𝑏𝑛−1,𝑝𝑛) (20)

𝑑(𝑏𝑛,𝑝𝑛+1) =𝑑�𝐹(𝑏𝑛−1,𝑞𝑛−1),𝐹(𝑝𝑛,𝑎𝑛)�,

≤ 𝛽1�𝑑(𝑏𝑛−1,𝑝𝑛)�𝑑(𝑏𝑛−1,𝑝𝑛) +𝛽2�𝑑(𝑎𝑛,𝑞𝑛−1)�𝑑(𝑎𝑛,𝑞𝑛−1) (21)

𝑑(𝑏𝑛+1,𝑝𝑛) =𝑑�𝐹(𝑏𝑛,𝑞𝑛),𝐹(𝑝𝑛−1,𝑎𝑛−1)�,

≤ 𝛽1�𝑑(𝑏𝑛,𝑝𝑛−1)�𝑑(𝑏𝑛,𝑝𝑛−1) +𝛽2�𝑑(𝑎𝑛−1,𝑞𝑛)�𝑑(𝑎𝑛−1,𝑞𝑛) (22) for all 𝑛 ∈ 𝑁+. Let

𝑒𝑛=𝑑(𝑎𝑛,𝑞𝑛+1) +𝑑(𝑏𝑛,𝑝𝑛+1)

for all 𝑛 ∈ 𝑁+. Using (19), (20), (21) and (22), we get

𝑒𝑛=𝑑(𝑎𝑛,𝑞𝑛+1) +𝑑(𝑏𝑛,𝑝𝑛+1)

≤ 𝛽1�𝑑(𝑎𝑛−1,𝑞𝑛)�𝑑(𝑎𝑛−1,𝑞𝑛) +𝛽2�𝑑(𝑏𝑛−1,𝑝𝑛)�𝑑(𝑏𝑛−1,𝑝𝑛)

+ 𝛽1�𝑑(𝑏𝑛−1,𝑝𝑛)�𝑑(𝑏𝑛−1,𝑝𝑛) +𝛽2�𝑑(𝑎𝑛−1,𝑞𝑛)�𝑑(𝑎𝑛−1,𝑞𝑛)

= (𝛽1 + 𝛽2 )��𝑑(𝑎𝑛−1,𝑞𝑛) +𝑑(𝑏𝑛−1,𝑝𝑛)�� �𝑑(𝑎𝑛−1,𝑞𝑛) +𝑑(𝑏𝑛−1,𝑝𝑛)�

<𝑒𝑛−1.

and

𝑠𝑛=𝑑(𝑎_(𝑛+ 1) ,𝑞_(𝑛) ) +𝑑(𝑏_𝑛,𝑝_(𝑛+ 1) )

≤ 𝛽1�𝑑(𝑎𝑛,𝑞𝑛−1)�𝑑(𝑎𝑛,𝑞𝑛−1) +𝛽2�𝑑(𝑏𝑛−1,𝑝𝑛)�𝑑(𝑏𝑛−1,𝑝𝑛) + 𝛽1�𝑑(𝑏𝑛−1,𝑝𝑛)�𝑑(𝑏𝑛−1,𝑝𝑛) +𝛽2�𝑑(𝑎𝑛,𝑞𝑛−1)�𝑑(𝑎𝑛,𝑞𝑛−1) = (𝛽1 + 𝛽2)�𝑑(𝑎𝑛,𝑞𝑛−1) +𝑑(𝑏𝑛−1,𝑝𝑛)��𝑑(𝑎𝑛,𝑞𝑛−1) +𝑑(𝑏𝑛−1,𝑝𝑛)�

<𝑠𝑛−1 .

Then we get

0≤ 𝑒𝑛𝑒𝑛−1≤ 𝑒𝑛−2≤ ⋯ 𝑒0. (23)

and

0≤ 𝑒𝑛≤ 𝑒𝑛−1 ≤ 𝑒𝑛−2≤ ⋯ 𝑒0. (24)

𝑑(𝑎𝑛,𝑝𝑛) =𝑑�𝐹(𝑎𝑛−1,𝑝𝑛−1),𝐹(𝑞𝑛−1,𝑏𝑛−1)�,

≤ 𝛽1�𝑑(𝑎𝑛−1,𝑞𝑛−1)�𝑑(𝑎𝑛−1,𝑞𝑛−1) +𝛽2�𝑑(𝑏𝑛−1,𝑝𝑛−1)�𝑑(𝑏𝑛−1,𝑝𝑛−1) (25)

and

𝑑(𝑏𝑛,𝑞𝑛) =𝑑�𝐹(𝑏𝑛−1,𝑝𝑛−1),𝐹(𝑝𝑛−1,𝑎𝑛−1)�,

≤ 𝛽1�𝑑(𝑏𝑛−1,𝑝𝑛−1)�𝑑(𝑏𝑛−1,𝑝𝑛−1) +𝛽2�𝑑(𝑎𝑛−1,𝑞𝑛−1)�𝑑(𝑎𝑛−1,𝑞𝑛−1) (26)

for all 𝑛 ∈ 𝑁+. Let

𝑡𝑛 = 𝑑(𝑎𝑛,𝑞𝑛) + 𝑑(𝑏𝑛,𝑝𝑛)

for all 𝑛 ∈ 𝑁+. Combining (25) and (26), we observe that

𝑡𝑛=𝑑(𝑎𝑛, 𝑞𝑛) + 𝑑(𝑏𝑛, 𝑝𝑛)

≤ 𝛽1�𝑑(𝑎𝑛−1, 𝑞𝑛−1)�𝑑(𝑎𝑛−1 , 𝑞𝑛−1) + 𝛽2�𝑑(𝑏𝑛−1, 𝑝𝑛−1)�𝑑(𝑏𝑛−1, 𝑝𝑛−1) + 𝛽1�𝑑(𝑏𝑛−1, 𝑝𝑛−1)�𝑑(𝑏𝑛−1, 𝑝𝑛−1) + 𝛽2�𝑑(𝑎𝑛−1, 𝑞𝑛−1)�𝑑(𝑎𝑛−1, 𝑞𝑛−1)

= (𝛽1+ 𝛽2)�𝑑(𝑎𝑛−1, 𝑞𝑛−1) + 𝑑(𝑏𝑛−1, 𝑝𝑛−1)��𝑑(𝑎𝑛−1, 𝑞𝑛−1) + 𝑑(𝑏𝑛−1, 𝑝𝑛−1)�

<𝑡𝑛−1 .

So we get

0≤ 𝑡𝑛≤ 𝑡𝑛−1≤ 𝑡𝑛−2≤ ⋯ ≤ 𝑡0. (27)

We obtain that

𝑑(𝑎𝑛, 𝑞𝑚)≤ 𝑑(𝑎𝑛, 𝑞𝑛+1) + 𝑑(𝑎𝑛+1, 𝑞𝑛+1) +⋯+ 𝑑(𝑎𝑚−1,𝑞𝑚),

𝑑(𝑏𝑛, 𝑝𝑚)≤ 𝑑(𝑏𝑛, 𝑝𝑛+1) + 𝑑(𝑏𝑛+1, 𝑝𝑛+1) +⋯ + 𝑑(𝑏𝑚−1, 𝑝𝑚), 𝑑(𝑎𝑚, 𝑞𝑛)≤ 𝑑(𝑎𝑚, 𝑞𝑚−1) +𝑑(𝑎𝑚−1, 𝑞𝑚−1) +⋯ + 𝑑(𝑎𝑛+1, 𝑞𝑛),

for each 𝑛,𝑚 ∈ 𝑁, 𝑛 < 𝑚. Thus, from (23), (24) (27) and (28) , we have

𝑑(𝑎𝑛, 𝑞𝑚) + 𝑑(𝑏𝑚, 𝑝𝑛)≤ �𝑑(𝑎𝑛, 𝑞𝑛+1) + 𝑑(𝑏𝑛+1, 𝑝𝑛)�

+�𝑑(𝑎𝑛+1, 𝑞𝑛+1) + 𝑑(𝑏𝑛+1, 𝑝𝑛+1)�+⋯

+�𝑑(𝑎𝑚−1, 𝑞𝑚−1) + 𝑑(𝑏𝑚−1, 𝑝𝑚−1)�

+�𝑑(𝑎𝑚−1, 𝑞𝑚) + 𝑑(𝑏𝑚, 𝑝𝑚−1)�,

=𝑒𝑛+ 𝑡𝑛+1 + 𝑒𝑛+1 +⋯ + 𝑡𝑚−1 + 𝑒𝑚−1 ,

<𝑒0+𝑡0 (29)

and

𝑑(𝑎𝑚, 𝑞𝑛) + 𝑑(𝑏𝑛, 𝑝𝑚)≤ �𝑑(𝑎𝑚, 𝑞𝑚−1) +𝑑(𝑏𝑚−1, 𝑝𝑚)�

+�𝑑(𝑎𝑚−1, 𝑞𝑚−1) +𝑑(𝑏𝑚−1, 𝑝𝑚−1)�+⋯

+�𝑑(𝑎𝑛+1, 𝑞𝑛+1) +𝑑(𝑏𝑛+1, 𝑝𝑛+1)�

+�𝑑(𝑎𝑛+1, 𝑞𝑛) +𝑑(𝑏𝑛, 𝑝𝑛+1)�,

= 𝑠𝑚−1+ 𝑡𝑚−1+⋯ + 𝑠𝑛+1 + 𝑡𝑛+1 + 𝑠𝑛,

≤ 𝑒0+ 𝑡0 (30)

for 𝑛< 𝑚. Since, for an arbitrary 𝜖> 0, there exists 𝑛0 such that 𝑒0+ 𝑡0 < 𝜖

3 and 𝑠0 + 𝑡0< 𝜖

3 from (13) and

(14) we have

𝑑(𝑎𝑛,𝑞𝑚) + 𝑑(𝑏𝑚,𝑝𝑛) < 𝜖₃

for each 𝑛,𝑚 ≥ 𝑛₀. Then (𝑎_𝑛,𝑝_𝑛) and (𝑏_𝑛,𝑞_𝑛) are Cauchy bisequences. Because of completeness of (𝑋,𝑌,𝑑), there exist 𝑎,𝑏 ∈ 𝑋 and 𝑝,𝑞 ∈ 𝑌 with

𝑙𝑖𝑚𝑛→∞𝑎𝑛 = 𝑞 ,𝑙𝑖𝑚𝑛→∞𝑏𝑛 = 𝑝 ,𝑙𝑖𝑚𝑛→∞𝑝𝑛 = 𝑏, 𝑎𝑛𝑑𝑙𝑖𝑚𝑛→∞𝑞𝑛= 𝑎. (31)

Then there exists 𝑛₁∈ 𝑁 with

𝑑(𝑎𝑛,𝑞) <𝜖₃, 𝑑(𝑏𝑛,𝑝) < 𝜖₃, 𝑑(𝑏,𝑝𝑛) <𝜖₃ 𝑎𝑛𝑑 𝑑(𝑎,𝑞_𝑛) < 𝜖₃

for all 𝑛 ≥ 𝑛₁ and every 𝜖> 0. Since (𝑎_𝑛,𝑞_𝑛) and (𝑏_𝑛,𝑝_𝑛) are Cauchy bisequences, we get

𝑑(𝑎𝑛,𝑞𝑛) <𝜖₃and 𝑑(𝑏𝑛,𝑝𝑛) <𝜖₃ .

So, from (18), we have

𝑑(𝐹(𝑎,𝑝),𝑞)≤ 𝑑(𝐹(𝑎,𝑝),𝑞𝑛+1) +𝑑(𝑎𝑛+1,𝑞𝑛+1) +𝑑(𝑎𝑛+1,𝑞)

=𝑑�𝐹(𝑎,𝑞),𝐹(𝑝𝑛,𝑏𝑛)� +𝑑(𝑎𝑛+1,𝑞𝑛+1) +𝑑(𝑎𝑛+1,𝑞)

≤ 𝛽1�𝑑(𝑎,𝑞𝑛)�𝑑(𝑎,𝑞𝑛) +𝛽2�𝑑(𝑏,𝑝𝑛)�𝑑(𝑏,𝑝𝑛) +𝑑(𝑎𝑛+1,𝑞𝑛+1) +𝑑(𝑎𝑛+1,𝑞)

<𝛽1�₃𝜖�_{3 +}𝜖 𝛽2 �𝜖₃�𝜖_{3 +} 𝜖_{3 +}𝜖₃

=𝜖_{3 + 23} 𝜖 <𝜖

for each 𝑛 ∈ 𝑁and 𝜆 < 1. Then 𝑑(𝐹(𝑎,𝑝),𝑞) = 0. Hence, 𝐹(𝑎,𝑝) =𝑞. Similarly, we get

𝑑(𝑎,𝑞) =𝑑( 𝑙𝑖𝑚𝑛→∞𝑞𝑛,𝑙𝑖𝑚𝑛→∞𝑎𝑛) =𝑙𝑖𝑚𝑛→∞𝑑(𝑎𝑛,𝑞𝑛) = 0

and

𝑑(𝑏,𝑝) =𝑑( 𝑙𝑖𝑚𝑛→∞𝑝𝑛, 𝑙𝑖𝑚𝑛→∞𝑏𝑛) =𝑙𝑖𝑚𝑛→∞𝑑(𝑏𝑛,𝑞𝑛) = 0.

So, a = q and b = p. Therefore, (𝑎,𝑝)∈ 𝑋2∩ 𝑌2 is a coupled fixed point of F. As in the proof of the Theorem 6, uniqueness of the coupled fixed point of F can be shown easily.

In the setting of 𝛽1(𝑡) and 𝛽₂(𝑡) if we take 𝛽1(𝑡) =𝑘 and 𝛽2 (𝑡) =𝑙 in Theorem 6.

Corollary 11: Let (𝑋, 𝑌, 𝑑) be a complete bipolar metric space, 𝐹 ∶ (𝑋× 𝑌, 𝑌 × 𝑋)→ (𝑋, 𝑌) be a covariant mapping and k, l be non-negative constants. If the condition

𝑑�𝐹(𝑎, 𝑝), 𝐹(𝑞, 𝑏)� ≤ 𝑘𝑑(𝑎, 𝑞) + 𝑙𝑑(𝑏, 𝑝), 𝑘+𝑙 < 1

holds for all 𝑎, 𝑏 ∈ 𝑋,𝑝, 𝑞 ∈ 𝑌 , then 𝐹 ∶ (𝑋×𝑌 )∪ (𝑌× 𝑋)\𝑡𝑜 𝑋 ∪ 𝑌 has a unique coupled fixed point.

Corollary 12: Let (X, Y, d) be a complete bipolar metric space.𝐹 ∶ (𝑋 \𝑡𝑖𝑚𝑒𝑠𝑌,𝑌 \𝑡𝑖𝑚𝑒𝑠𝑋)→(𝑋,𝑌) be a covariant mapping and k, l be non-negative constants. If the condition

𝑑�𝐹(𝑎,𝑝),𝐹(𝑞,𝑏)� ≤ _{2 (}𝑘 𝑑(𝑎,𝑞) + 𝑑(𝑏,𝑝)), 𝑘< 1

Example 13: Let 𝑈_𝑛(𝑅 ) and 𝐿_𝑛(𝑅) be the sets of all 𝑛× 𝑛 upper and lower triangular matrices over 𝑅, respectively. A function 𝑑 ∶ 𝑈_𝑛(𝑅) × 𝐿_𝑛(𝑅)→ 𝑅+ be defined as

𝑑(𝐴,𝐵) =Σ𝑛𝑖,𝑗=1 |𝑎𝑖𝑗 − 𝑏𝑖𝑗 |

for all 𝐴 = �𝑎𝑖𝑗�

𝑛× 𝑛 ∈ 𝑈𝑛(𝑅) and 𝐵 = �𝑏𝑖𝑗�𝑛× 𝑛∈ 𝐿𝑛(𝑅). Then it is apparent that (𝑈𝑛(𝑅),𝐿𝑛(𝑅), 𝑑) is a

complete bipolar metric space. We take a covariant mapping 𝐹 ∶(𝑈_𝑛(𝑅)2,𝐿_𝑛(𝑅)2)→ (𝑈_𝑛(𝑅),𝐿_𝑛(𝑅)) such as

𝐹(𝐴,𝐵) =�_2𝑟𝑟_{+ 1} 𝑎𝑖𝑗+_3𝑟𝑟_{+ 1} 𝑏𝑖𝑗 � 𝑛× 𝑛

where�𝐴 = �𝑎𝑖𝑗�

𝑛× 𝑛 , 𝐵 = �𝑏𝑖𝑗�𝑛× 𝑛� ∈ 𝑈𝑛(𝑅)2∪ 𝐿𝑛(𝑅)2.

Then we get

𝑑�𝐹(𝐴,𝐵), 𝐹(𝐶,𝐷)�=𝑑 � �𝑎𝑖𝑗+𝑏𝑖𝑗

3 �_{𝑛× 𝑛} , � 𝑐𝑖𝑗+𝑑𝑖𝑗

3 �_{𝑛× 𝑛} �

= Σ𝑖𝑛,𝑗=1 |�_(2𝑟+1𝑟 ₎𝑎𝑖𝑗 + _3𝑟+1𝑟 𝑏𝑖𝑗 −_2𝑟+1𝑟 𝑐𝑖𝑗 −_3𝑟+1𝑟 𝑑𝑖𝑗) �

≤ Σ𝑖𝑛,𝑗=1��_2𝑟+1𝑟 � ��𝑎𝑖𝑗−𝑐₃ 𝑖𝑗�� +�_3𝑟+1𝑟 � ��𝑏𝑖𝑗−𝑑₃ 𝑖𝑗�� �

=�_2𝑟𝑟_{+ 1}� 𝑑(𝐴,𝐶) +�_3𝑟𝑟_{+ 1} � 𝑑(𝐵,𝐷)

for all 𝐴 = �𝑎𝑖𝑗�

𝑛× 𝑛 ,𝐵 = �𝑏𝑖𝑗�𝑛× 𝑛 ∈ 𝑈𝑛(𝑅) and 𝐶 = �𝑐𝑖𝑗�𝑛× 𝑛,𝐷 = �𝑑𝑖𝑗�𝑛× 𝑛 ∈ 𝐿𝑛(𝑅).

Therefore, the equation (17) is satisfied for 𝛽1(𝑡) =� 𝑟

2𝑟+1� 𝑎𝑛𝑑𝛽2(𝑡) = � 𝑟

3𝑟+1 � . Then from Corollary 7, F has a

unique coupled fixed point. It is obvious that the coupled fixed point is (0_{𝑛× 𝑛} , 0_{𝑛× 𝑛})∈ 𝑈_𝑛(𝑅)∩ 𝐿_𝑛(𝑅) where 0_{𝑛× 𝑛} is the null matrix.

On the other hand, if

𝐹 ∶(𝑈𝑛(𝑅)2,𝐿𝑛(𝑅)2)→ �𝑈𝑛(𝑅),𝐿𝑛(𝑅)�

is defined by 𝐹(𝐴,𝐵) = �𝑎𝑖𝑗+𝑏𝑖𝑗

3 �𝑛× 𝑛

where �𝐴=�𝑎𝑖𝑗�

𝑛× 𝑛, 𝐵=�𝑏𝑖𝑗�𝑛× 𝑛� ∈ 𝑈𝑛(𝑅)2∪ 𝐿𝑛(𝑅)2. Then it can be observed that �_(2𝑟 (𝑟_{+ 1)} ) � 𝑑(𝐴,𝐶) +�_3𝑟𝑟_{+ 1} � 𝑑(𝐵,𝐷).

Then F satisfies the equation (17) for k = 1. Therefore, coupled fixed points of F are both (0_{𝑛× 𝑛}, 0_{𝑛× 𝑛})∈ 𝑈𝑛(𝑅)∩

𝐿𝑛(𝑅) and (𝐼𝑛, 𝐼𝑛)∈ 𝑈𝑛( 𝑅)∩ 𝐿𝑛( 𝑅) where 0𝑛× 𝑛 is the null matrix and 𝐼𝑛 is the identity matrix. As it can be seen

from this expression, F has not a unique coupled fixed point. Thus, the conditions 𝛽₁(𝑡) +𝛽₂(𝑡) < 1 where (𝛽₁(𝑡) =𝑘

𝑎𝑛𝑑𝛽2 (𝑡) = 𝑙) in Corollary 7 and 𝛽1(𝑡) +𝛽2(𝑡) < 1 in Theorem 6 are the most appropriate conditions for satisfying

the uniqueness of coupled fixed point.

Example 14: Let 𝑈_𝑛( 𝑅) and 𝐿_𝑛( 𝑅) be the sets of all 𝑛× 𝑛 upper and lower triangular matrices over R, respectively. A function 𝑑 ∶ 𝑈_𝑛( 𝑅) × 𝐿_𝑛( 𝑅)→ 𝑅+ be defined as

𝑑(𝐴,𝐵) =Σ𝑛𝑖,𝑗=1�𝑎𝑖𝑗 − 𝑏𝑖𝑗 �

for all 𝐴 = �𝑎𝑖𝑗�

𝑛× 𝑛∈ 𝑈𝑛( 𝑅) and 𝐵=�𝑏𝑖𝑗�𝑛× 𝑛 ∈ 𝐿𝑛( 𝑅). Then it is apparent that (𝑈𝑛( 𝑅),𝐿𝑛(𝑅), 𝑑) is a

complete bipolar metric space. We take a covariant mapping 𝐹 ∶(𝑈_𝑛( 𝑅)2,𝐿_𝑛( 𝑅)2)→ �𝑈𝑛( 𝑅),𝐿_𝑛( 𝑅)� such as

𝐹(𝐴,𝐵) =�𝑎𝑖𝑗+𝑏𝑖𝑗

3 �_{𝑛× 𝑛} 𝑤ℎ𝑒𝑟𝑒 �𝐴 = �𝑎𝑖𝑗�_{𝑛× 𝑛} , 𝐵 = �𝑏𝑖𝑗�_{𝑛× 𝑛}� ∈ 𝑈𝑛( 𝑅)2_{∪ 𝐿𝑛( 𝑅)}2_.

Then we get

𝑑�𝐹(𝐴,𝐵), 𝐹(𝐶,𝐷)�=𝑑 � �𝑎𝑖𝑗+₃ 𝑏𝑖𝑗� 𝑛× 𝑛

, �𝑐𝑖𝑗+₃𝑑𝑖𝑗� 𝑛× 𝑛

�

= Σ𝑖𝑛,𝑗=1 �𝑎𝑖𝑗 +𝑏𝑖𝑗 − 𝑐𝑖𝑗₃ − 𝑑𝑖𝑗 �

≤ Σ𝑖𝑛,𝑗=1 ��𝑎𝑖𝑗− 𝑐𝑖𝑗₃ �+ �𝑏𝑖𝑗− 𝑑𝑖𝑗₃ ��

= 1_{3 (}𝑑(𝐴,𝐶) + 𝑑(𝐵,𝐷))

for all 𝐴 = �𝑎𝑖𝑗�

𝑛× 𝑛 ,𝐵 = �𝑏𝑖𝑗�𝑛× 𝑛 ∈ 𝑈𝑛( 𝑅) and 𝐶 = �𝑐𝑖𝑗�𝑛×𝑛 ,𝐷 = �𝑑𝑖𝑗�𝑛× 𝑛 ∈ 𝐿𝑛( 𝑅).

Therefore, the equation (17) is satisfied for 𝑘 = 2

3 . Then from Corollary 8, F has a unique coupled fixed point. It is

On the other hand, if 𝐹 ∶(𝑈_𝑛( 𝑅)2,𝐿_𝑛( 𝑅)2)→ (𝑈_𝑛( 𝑅),𝐿_𝑛( 𝑅)) is defined by𝐹(𝐴,𝐵) = �𝑎𝑖𝑗+𝑏𝑖𝑗

3 �_{𝑛× 𝑛}

where �𝐴 = �𝑎𝑖𝑗�

𝑛× 𝑛 , 𝐵 = �𝑏𝑖𝑗�𝑛× 𝑛� ∈ 𝑈𝑛(𝑅)2∪ 𝐿𝑛(𝑅)2. Then it can be observed that 𝑑�𝐹(𝐴,𝐵), 𝐹(𝐶,𝐷)� ≤ ₂1 �𝑑(𝐴,𝐶) +𝑑(𝐵,𝐷)�.

Then F satisfies the equation (17) for k = 1. Therefore, coupled fixed points of F are both (0_{𝑛× 𝑛}, 0_{𝑛× 𝑛})∈ 𝑈_𝑛(𝑅)∩

𝐿𝑛(𝑅)𝑎𝑛𝑑(𝐼𝑛, 𝐼𝑛)∈ 𝑈𝑛(𝑅)∩ 𝐿𝑛(𝑅) where 0𝑛× 𝑛 is the null matrix and 𝐼𝑛 is the identity matrix. As it can be seen

from this expression, F has not a unique coupled fixed point. Thus, the conditions k < 1 in Corollary 8 and k + l < 1 in Theorem 6 are the most appropriate conditions for satisfying the uniqueness of coupled fixed point.

ACKNOWLEDGMENT

The authors express deep gratitude to the chief editor and referees for valuable comments and suggestions.

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